Problem 88 - College Algebra

Question

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Step-by-Step Solution

Answer

A logarithmic equation \( \log_b(a) = c \) and an exponential equation \( b^c = a \) represent the same relationship between \( a \), \( b \) and \( c \). The two forms are simply different ways of expressing the same mathematical truth. The base and result in a logarithm become the base and outcome in an exponential equation, respectively, with the exponent (in exponential form) being the result in the logarithmic form.

Step 1: Understanding Logarithmic and Exponential Forms

First, it's important to grasp the basic forms of these two types of equations. A logarithmic equation is generally given in the form: \( \log_b(a) = c \), where \( b \) is the base, \( a \) is the argument, and \( c \) is the result. On the other hand, an exponential equation is commonly given in the form: \( b^c = a \), where \( b \) is the base, \( c \) is the exponent, and \( a \) is the result.

Step 2: Converting Logarithmic Form to Exponential Form

To convert a logarithmic equation to an exponential equation, you can understand \( \log_b(a) = c \) to mean that 'the base \( b \) raised to the power \( c \) equals \( a \)' which gives us the exponential form \( b^c = a \). This interpretation comes from the definition of a logarithm.

Step 3: Converting Exponential Form to Logarithmic Form

Similarly, to translate an exponential equation to a logarithmic one, the equation \( b^c = a \) is seen as '\( c \) is the power to which the base \( b \) must be raised to get \( a \)' which provides us with the logarithmic form \( \log_b(a) = c \). Again, this interpretation arises from the definition of an exponential function.